An Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes

Transcription

1 45th AIAA Aerospace Sciences Meeting and Exhibit, January 8, 27, Reno, NV An Unsteady Discrete Adjo Formulation for wo-dimensional Flow Problems with Deforming Meshes Karthik Mani Department of Mechanical Engineering, University of Wyoming, Laramie, Wyoming Dimitri J. Mavriplis Department of Mechanical Engineering, University of Wyoming, Laramie, Wyoming A method to apply the discrete adjo for computing sensitivity derivatives in two-dimensional unsteady flow problems is presented. he approach is to first develop a forward or tangent linearization of the nonlinear flow problem where each individual component building up the complete flow solution is differentiated against the design variables using the chain rule. he reverse or adjo linearization is then constructed by transposing and reversing the order of multiplication of the forward problem. he developed algorithm is very general in that it applies directly to the Arbitrary-Lagrangian-Eulerian (ALE) form of the governing equations and includes the effect of deforming meshes in unsteady flows. It is shown that an unsteady adjo formulation is essentially a single backward egration in time, and that the cost of constructing the final sensitivity vector is close to that of solving the unsteady flow problem egrated forward in time. It is also shown that the unsteady adjo formulation can be applied to time-egration schemes of different orders of accuracy with minimal changes to the base formulation. he developed technique is then applied to three optimization examples, the first where the shape of a pitching airfoil is morphed to match a target time-dependent load profile, the second where the shape is optimized to match a target time-dependent pressure profile, and the last where the time-dependent drag profile is minimized without any loss in lift. I. Introduction In the recent past, the use of adjo equations has become a popular approach for solving aerodynamic design optimization problems based on computational fluid dynamics. 6 Adjo equations are a very powerful tool in the sense that they allow the computation of sensitivity derivatives of an objective function to a set of given inputs at a cost which is essentially independent of the number of inputs. his is in contrast to the brute-force finite-difference method, where each input or design variable has to perturbed individually to obtain a corresponding effect on the output. his is a tedious and costly process which is of little use when there a large number of design variables or inputs. Adjo equations may be utilized in two forms, either continuous or discrete. In the continuous approach, the governing non-linear flow equations are differentiated with respect to the design variables and then discretized before obtaining a solution, whereas in the case of the discrete adjo approach, this order is reversed. In the discrete approach the governing equations are discretized and the discrete version then differentiated against the design variables. heoretically, both methods converge to the same solution as the limit of the discretization resolution tends to infinity. he continuous adjo approach offers significant flexibility in the discretization process, however, the discrete adjo approach provides exact sensitivity derivatives of the original discretized governing equations. Also, the boundary conditions in the case of the discrete adjo are straightforward derivatives of existing boundary conditions for the governing flow equations. Another significant advantage in using the discrete approach comes from the fact that the governing flow equations in many cases are solved implicitly, thus requiring a linearization. his linearization is therefore readily available for use in computing the discrete adjo. he use of adjo equations either in continuous or discrete form for solving steady-state aerodynamic optimization problems 4, 7 is now fairly well established. However, relatively little work has been done in applying these methods for unsteady time-dependent problems., 2 In the context of unsteady flows, frequency domain methods3, 4 Graduate Student, AIAA Member; kmani@uwyo.edu. Professor, AIAA Associate Fellow; mavripl@uwyo.edu. of 34

2 have been investigated that allow for reduced computational expense particularly for problems with strong periodic behavior. he goal of this paper is to develop an efficient framework for the computation of sensitivities for unsteady flow problems operating directly in the time domain. Although applicable to any type of unsteady problem, the method may be more costly when compared with the frequency domain approach. However, both methods are complimentary to one and other. Most importantly the developed algorithm takes o account the effect of deforming meshes which are becoming commonplace in unsteady computations. Although rigid-body motion in unsteady computations may be handled by overset-grid methods, shape deformation in such flows can be handled only via the use of mesh deformation equations. his is especially true in the field of aeroelasticity. Ultimately the procedure may be extended to applications such as unsteady adaptive mesh refinement built on current adjo based mesh refinement in steady problems. 5, 6 here is also potential in using the core of the algorithm in time-step size adaptation for unsteady flows. II. Analysis Problem Formulation A. Governing Equations of the Flow Problem in ALE Form he conservative form of the Euler equations is used in solving the flow problem. he paper is limited to inviscid flow problems since the primary focus is the development and validation of an unsteady adjo method. Extension of the algorithm to viscous flow problems with the inclusion of turbulence models should prove to be straightforward. he differences arise only in the linearization of the additional flux contributions and not in the base formulation presented in the paper. In vectorial form the conservative form of the Euler equations may be written as: U(x, t) t + F(U) = () Applying the divergence theorem and egrating over a moving control volume A(t) yields: Using the differential identity: equation (2) is rewritten as: Z A(t) or when considering cell-averaged values for the state U as: U t ZdB(t) da + F(U) ndb = (2) Z Z U UdA = t A(t) A(t) t ZdB(t) da + (ẋ n)db (3) Z Z UdA + [F(U) ẋu] ndb = (4) t A(t) db(t) AU t Z + [F(U) ẋu] ndb = (5) db(t) his is the Arbitrary-Lagrangian-Eulerian (ALE) finite-volume form of the Euler equations. he equations are required in ALE form since the problem involves deforming meshes where mesh elements change in shape and size at each time-step. Here A refers to the area of the control volume, ẋ is the vector of mesh face or edge velocities, and n is the unit normal of the face or edge. he state vector U of conserved variables and the cartesian inviscid flux vector F = (F x,f y ) are: U = ρ ρu ρv E t, Fx = ρu ρu 2 + p ρuv u(e t + p), Fy = ρv ρuv ρv 2 + p v(e t + p), (6) Here ρ is the fluid density, (u,v) are the cartesian fluid velocity components, p is the pressure and E t is the total energy. For an ideal gas, the equation of state relates the pressure to total energy by: [ p = (γ ) E t ] 2 ρ(u2 + v 2 ) (7) where γ =.4 is the ratio of specific heats. 2 of 34

3 B. emporal Discretization he time derivative term in the Euler equations is discretized using both a first-order accurate backward-differenceformula (BDF), and a second-order accurate BDF2 scheme. hese discretizations are shown in equations (8) and (9) respectively. he index n is used to indicate the current time-level as the convention throughout the paper. he discretization of the BDF2 scheme shown in equation (9) is based on a uniform time-step size. AU t AU t = An U n A n U n t 3 2 = An U n 2A n U n + 2 An 2 U n 2 t (8) (9) C. Spatial Discretization he solver uses a cell-centered finite-volume formulation where the inviscid flux egral around a closed control volume is discretized as: Z db(t) n edge [F(U) ẋu] ndb = S cell = i= F i B i V ei U cell B i () where B i is the edge length,v ei is the normal edge velocity, and Fi is the normal flux across the edge. he normal flux across the edge is computed as: F edge = 2 {F L (q L ) + F R(q R ) + [Â](q L q R )} () where q L, q R are the left and right extrapolated state vectors and F L, F R edge computed as: ρv F ρv = u + ˆn x p ρv v + ˆn y p (E t + p)v are the left and right normal fluxes for the (2) he velocity normal to the edge V is defined as u ˆn x + v ˆn y, where ˆn x and ˆn y are the unit edge normal vector components. he dissipative flux is computed using Roe s approximate Riemann solver 7 and therefore matrix [Â] is the Roe averaged flux Jacobian. Second-order spatial accuracy is achieved by using a gradient based reconstruction scheme to determine the state vector at the edges based on cell-centered values. he extrapolated state q is calculated as: q = U + U r (3) Here r is the vector between the cell center and the edge center, and U is the gradient of the state vector. he gradient is computed by first obtaining nodal state values via arithmetic averages of cell-centered values from elements that share the node, and then using Green-Gauss reconstruction to obtain the final state gradients at the cell-centers. An arithmetic average is used rather than more accurate polynomial based erpolation methods in order simplify the linearization procedure. D. he Discrete Geometric Conservation Law (GCL) he discrete geometric conservation law (GCL) requires that a uniform flow field be preserved when equation (5) is egrated in time. In other words the deformation of the computational mesh should not roduce conservation errors in the solution of the flow problem. his translates o U =constant being an exact solution of equation (5). For a conservative scheme, the egral of the inviscid fluxes around a closed contour goes to zero when U =constant. Applying these conditions to equation (5) results in the mathematical description of the GCL as stated below. A t ZdB(t) ẋ ndb = (4) 3 of 34

4 Equation (4) implies that the change in area of a control volume should be discretely equal to the area swept by the boundary of the control volume. he vector of cartesian edge velocities ẋ for each of the edges encompassing the control volume must therefore be chosen such that equation (4) is satisfied. he dot product of the cartesian edge velocities and the edge normal vector is represented by the term V e. here are several methods to compute the edge velocities while satisfying the GCL, but in previous work it has been shown that satisfaction of the GCL is not a sufficient condition to achieve the underlying accuracy of the chosen time-egration scheme. 8, 9 he GCL has been specifically derived for first, second and third-order backward difference (BDF) time-egration schemes and also for the fourth-order accurate IRK64 implicit Runge-Kutta scheme. 9 he form of the GCL described in Ref. 9 is used exclusively in this work. E. Mesh Deformation Strategy Deformation of the mesh is achieved through the linear tension spring analogy which approximates the mesh as a network of er-connected springs. he spring coefficient is assumed to be inversely proportional to the edge length. wo independent force balance equations are formulated for each node based on displacements of neighbors. his results in a nearest neighbor stencil for the final linear system to be solved. he linear system that relates the erior vertex displacements in the mesh to known displacements on the boundaries is: [K]δx = δx sur f (5) he stiffness matrix [K] is a sparse block matrix that can be treated as a constant since it is based on the initial configuration of the mesh and remains unchanged through the time-egration process. For each node in the mesh, the displacement can be solved as δx i = δx sur f i n j j= [O] i jδx j [D] i (6) where the off-diagonal term [O] i j and diagonal term [D] i are 2 2 diagonal matrices as follows: [O] i j = k i j [I] for i j (7) [D] i = n j j= k i j [I] (8) where k i j refers to the stiffness of the corresponding edge. he mesh motion equations are elliptic in nature and can be solved using conventional smoothers such as Gauss-Seidel or Gauss-Jacobi iterations. he convergence of the system is a function of the mesh resolution and becomes prohibitively expensive as the mesh resolution becomes large. An agglomeration multigrid 2 approach is used to accelerate the convergence of the linear system. Being elliptic in nature the system is particularly well suited for acceleration using multigrid. F. Geometry Parametrization Modification of the baseline geometry is achieved through displacements of the surface nodes defining the geometry. In order to ensure smooth geometry shapes, any displacement of a surface node is controlled by a bump function which influences neighboring nodes with an effect that diminishes moving away from the displaced node. he bump function used for the results presented in this paper is the Hicks-Henne Sine bump function. 2 It is defined as: b i (x) = a sin 4 (πx m i ) (9) m i = ln(.5)/ln(x Mi ) (2) Here x Mi refers to the location where the bump is to be placed and is typically the x coordinate of the surface node which is to be displaced. b i (x) are the displacements of pos with coordinates ranging from to as a result of the displacement of the po at x Mi in accordance with the sine bump function. a is the magnitude of the bump placed at x Mi. he design variables for the examples presented in the paper are the magnitudes of bumps placed at surface nodes. he bump function is valid in the range of x,x Mi <, and is ideally suited for airfoil problems. For geometries extending beyond this range, the bump function may be mapped locally over some predetermined radius about the surface node of erest. he superposition of bump functions placed at each surface node is added to the shape of the baseline airfoil in order to deform its surface. Figure () shows a series of bumps with a magnitude of. placed between x <. 4 of 34

5 y x Figure. Hick-Henne Sine bump functions. III. Analysis Procedure he example problems chosen to demonstrate the unsteady adjo algorithm involves a deformable airfoil that pitches sinusoidally. he procedure to determine the unsteady flow solution for such a case is as follows:. Determine the surface coordinates (shape) of the deformed airfoil as δx base sur f = f (D + δd) (2) x sur f = xbase sur f + δxbase sur f (22) where δx base sur f are the displacements of the surface nodes computed as a function of the design variables D. In this particular case the functional operator refers to the Hicks-Henne Sine bump function described in the previous section. x base sur f refers to the surface coordinates of the baseline non-deformed airfoil. In the case of an optimization x base sur f refers to the airfoil from the previous design iteration rather than the original baseline configuration. 2. Solve mesh motion equations to determine displacements of erior nodes δx. Add to erior node coordinates of baseline mesh (or mesh from prior design iteration) to get the erior node distribution for the deformed airfoil as x = xbase sur f + δx. 3. Begin time-egration at time n =. 4. ransform the deformed surface coordinates to their new orientation for the current time-level n through multiplication of the rotation matrix [β] n which defines the prescribed motion of the pitching airfoil as a function of time. he center of rotation for the airfoil is taken to be the quarter-chord location. he rotation matrix [β] n is a function of the angle of attack at the current time-level and can be determined using the Sine function shown in equation (23). he transformed airfoil is then used to compute the surface displacement vector δx n sur f for the current time-level as shown in equation (25). α n = + α max sin(ωt n ) α o ( ) (23) [β] n = cosα n sinα n sinα n cosα n (24) δx n sur f = ([β] n [I])x sur f (25) 5. Solve the mesh motion equations again to obtain the erior node displacements δx n at time-level n. he erior node coordinates are then computed as x n = x + δxn, where x are the erior node coordinates for the deformed baseline airfoil at the time-level n =. 5 of 34

6 6. Calculate edge velocities V e in accordance with the GCL for the time-egration scheme used. For a BDF scheme these are functions of coordinates x n and xn and functions of coordinates x n, xn and x n 2 for the BDF2 scheme. Further details can be found in Ref. 9 New cell areas A n are also computed at this po using x n. 7. For an implicit time-step the non-linear flow residual is defined as: R n = (AU) t + S(V n e,n n e,u n ) = (26) where the discretization of the time derivative is based on the chosen time-egration scheme. he second term is the spatial residual of the flow and is a function of the edge velocities V e, the edge normals n e and the flow solution at the current time-level U n. 8. he non-linear flow residual is then solved using Newton s method as shown in equation (27) in conjunction with an agglomeration multigrid scheme to accelerate convergence of the ermediate linear systems. [ ] R(U k ) U k δu k = R(U k ) (27) U k+ = U k + δu k (28) δu k,u k+ = U n (29) 9. If the current time-level is n =, a steady-state solution is obtained for use as an initial condition by solving S(V n e,n n e,u n ) =.. he lift and drag coefficients CL n and Cn D are then computed for the current time-level. he load coefficients are functions of both the current flow solution and the current mesh coordinates. IV. Sensitivity Formulation A. Generalized Sensitivity Analysis Consider a non-linear operation that requires an input D and computes an output L. Mathematically this can be represented as L=F(D), where F is the non-linear operator acting on the input D. For example, in the case of a steady-state flow problem, the output L may be the drag acting on an airfoil, and the input D may be a vector of variables representing the shape of the airfoil. If the drag on the airfoil is to be minimized by changing the shape of the airfoil, the sensitivity of the drag to the shape is required. he output, which in this case is the drag is defined as the objective function L that requires minimization. he conventional method of determining the sensitivity dd dl is to obtain the output L for a baseline configuration with some known input D and then to perturb the shape of the baseline configuration by δd. his results in a change of the output L by δl, by which the sensitivity dd dl is approximated as δl/δd. his procedure describes the finite-difference method and in the case of multiple inputs and/or multiple outputs involves perturbing each input D and recording their effect on each L. It is obvious that the finite-difference procedure would require the numerical evaluation of the non-linear operator F for each input D in the problem. For multiple design variables and multiple outputs the sensitivities form a matrix of size n L by n D where n L is the number of objectives and n D is the number of design variables. he finite-difference method has been used effectively and extensively to develop the sensitivity matrix in many commercial optimization packages such as isight 22 and ModelCenter. 23 In order to determine dd dl it is possible to differentiate the analytical form of F(D) with respect to D. his describes the continuous approach. In the case of a flow problem based on the Euler equations it involves the analytical derivative of the Euler equations. he other option is to sequentially differentiate each discrete operation used to evaluate F(D) numerically. his is done by directly differentiating the code used to solve the discretized Euler equations which requires D as the input and computes L as the output. his is the discrete approach and is the focus of this paper. If the ermediate functional dependencies of the output L on the input D are written as: L(D) = L(F n (F n 2 (...F 2 (F (D))...)))) (3) then the sensitivity of L on D can be computed by differentiating the equation using the chain rule as: dl dd = L. F n... F 2. F F n F n 2 F (3) 6 of 34

7 For the case of a single input D and multiple outputs L, the sensitivity dd dl is a vector of size [n L,]. he product of the last two terms in equation (3) is either a matrix-vector multiplication or a vector-vector inner product that results in a vector. his implies that the final sensitivity dd dl can be computed as a series matrix-vector or vector-vector products. here are no expensive matrix-matrix multiplications required in this procedure. For cases with more than one input D, the series of multiplications must be performed for each input D. In situations with a single output L and multiple inputs D it becomes inefficient to use the forward linearization since the cost of obtaining the final sensitivity dl dd is directly proportional to the number of inputs D. his arises from the fact that the procedure has to be carried out for each input variable D. he problem is eliminated by transposing equation (3) to obtain a reverse linearization as shown below. dl dd = F. F 2 F. F n F n 2 he last term in equation (32) is now a vector of size [n L,], and the resulting sequence of multiplications again consist of either matrix-vector products or vector-vector inner products. In contrast with the forward procedure, for each additional output L the sequence of operations in equation (32) has to be repeated to obtain the matrix of sensitivities dl dd. o summarize, it is efficient to invoke the forward linearization for cases with multiple outputs L and a single or few inputs D, while the adjo or reverse linearization is more economical for cases with a single or few outputs L and multiple inputs D. B. Objective Function Formulation for the Unsteady Problem he objective function for the first unsteady optimization example is defined such that the time-dependent lift and drag profiles of the optimized airfoil match target or required time-dependent profiles. he objective function is based on the difference between the target and computed objective values at each time-level n. Figures (2(a)) and (2(b)) exemplify the formulation of the unsteady objective function. If CL n and Cn D refer to the lift and drag coefficients computed on the airfoil at time-level n during the pitch cycle, then the local objective at time-level n is defined as:... L F n (32) L n = (δc n L) 2 + (δc n D) 2 (33) δc n L = (C n L C n Ltarget) (34) δc n D = (C n D C n Dtarget) (35) where the factor of for the drag coefficient is roduced to equalize its order of magnitude with the lift coefficient. he global or time-egrated objective is taken to be the root-mean-square (RMS) average of the local objective functions: L g n=n f n= = Ln (36) n f + When the RMS error goes to zero, the target and computed load profiles match. discretization of a time-dependent objective L(t) defined as: his can be erpreted as the L(t) = δc L (t) + δc D (t) (37) For the second optimization example the local objectives are defined as the RMS difference between a target and computed pressure profile at each time-level, and the global or time-egrated objective is taken to be the arithmetic mean of the local objectives. It would be sufficient to target only the pressure profile at the end of the time-egration process since all other pressure profiles would automatically match up if any one of the profiles match. his arises from the fact that the pressure profile is a function of the airfoil shape and this shape remains fixed through the timeegration process for each design iteration. However, for the purpose of testing the algorithm and to study its behavior the entire set of time-dependent pressure distributions is used. L n = { i=n sur f i= [ p n i p i n target ] 2 n sur f } 2 (38) L g = n=n f n= Ln n f + (39) 7 of 34

8 CL CD n=4 profile n=5 n=3 n=6 n=2 n=9 δcd n n=8 n=7 n=6 n=5 n=4 n=3 n=7 n= n=8 n= n= n= n= n= n=2 δcl n n= n= arget profile arget profile n=9 profile α α (a) ime dependent lift profile (b) ime dependent drag profile Figure 2. ime dependent load profiles he third and last optimization example uses the same objective formulation as the first example with the target time-dependent lift profile set as that of the baseline configuration. he target time-dependent drag is taken to be zero since this allows for its reduction when the objective is minimized. Unlike in the case of a sinusoidally pitching airfoil, a more general unsteady problem of erest may not exhibit periodic behavior. In such cases, the objective function L is defined as some quantity of erest that is egrated in time or is based on flow variables and mesh coordinates at the final time-level n f. For example, such situations are commonplace in computational weather forecasting, 24 where some global quantity such as the average temperature or moisture in the future over some region is computed on the basis of known initial conditions from the present. his is a case where the quantity of erest is determined through egration of flow variables in time. he sensitivity of this quantity at the end of the time-egration cycle against the known initial conditions can then be computed using the unsteady adjo method. C. Sensitivity Formulation for the Unsteady Flow Problem he objective function L can be written with complete ermediate dependencies as a function of the design variables D in order to assist with the chain rule differentiation process. For the problem of the unsteady pitching airfoil the design variables are defined as a vector of weights controlling the magnitudes of bumps placed at surface node locations. For the purpose of deriving the unsteady adjo, the objective function definition of the first optimization example shown in equation (36) is used. he ermediate dependencies for a BDF based scheme can be written as follows: L g = F [ L,L,L 2...L n ] f (4) L n = F [CL,C n D] n (4) C n L,C n D = F [U n,n n e] (42) U n = F [ A n,a n,u n,s n] (43) S n = F [U n,v n e,n n e] (44) n n e = F [x n ] (45) A n = F [x n ] (46) V n e = F [ x n,x n ] (47) x n = F [ x,x n ] sur f (48) x n sur f = F [ x ] sur f (49) x sur f = F [D] (5) 8 of 34

9 where L g is the time-egrated objective function and the superscript n denotes values at the n th time-level. Sequentially differentiating equations (4) through (5) results in a general expression for the final sensitivity derivative dd dl for the unsteady flow problem: dl g n=n f dd = dl g [ L n U n n= dl n U n + Ln x n where several of the ermediate derivatives are not shown for simplicity. D. angent or Forward Linearization x n ] Consider the case of a single design variable D. Equation (26) and equation (5) form the constra equations for the sensitivity problem and can be differentiated with respect to the design variable D in order obtain convenient expressions for both the flow variable and the mesh-coordinate sensitivities. For a BDF time discretization the flow constra equation (i.e. the non-linear flow residual at a given time-level) can be written as: R n = An U n A n U n t (5) + S n (U n,v n e,n n e ) = (52) Based on the functional dependencies described previously, all terms appearing in the flow residual equation can be written in terms of the mesh coordinates and the flow state variables. In simplified functional form for a BDF time-egration scheme this is expressed as: Differentiating this with respect to the design variable D yields: dr n dd = Rn x n x n (53) R n = F [ x n,x n,u n,u n ] = (54) + Rn x n x n + Rn U n U n + Rn U n U n = (55) Rearranging the equation in terms of the flow Jacobian results in a convenient expression for the flow variable sensitivities. U n [ ] ( ) R n = R n x n U n x n + Rn x n x n + Rn U n U n (56) Substituting equation (56) o equation (5) yields the final form of the forward linearization of the flow problem. { [ dl g n=n ]( ) } f dd = dl g n= dl n Ln R n R n x n U n U n x n + Rn x n x n + Rn U n U n + Ln x n x n (57) he second constra is the set of mesh motion equations. he stiffness matrix [K] remains a constant and therefore does not affect the derivative. [K] xn = xn sur f x n = xn [K] sur f his can be substituted in place of the mesh derivative terms appearing in equation (57). he procedure to obtain the sensitivity dd dl using the forward linearization is as follows:. Begin time-egration loop at n = 2. Determine the derivative x sur f by differentiating the shape functions. 3. Multiply the surface node sensitivities with the rotation matrix [β] n to obtain the rotated surface node sensitivities for the current time-level. (58) (59) 9 of 34

10 4. Iteratively compute mesh sensitivity for current time-level as: 5. Compute flow residual sensitivity as: R n = ( R n x n [K] xn = xn sur f x n + Rn x n x n ) + Rn U n U n (6) (6) where the flow variable sensitivities and the mesh sensitivities for the previous time-step are known quantities. In the case of the first time-step these are the outputs from the steady-state sensitivity solution (if the unsteady solution is restarted from the steady solution) or these are derivatives of the initial conditions for the unsteady flow. he initial condition may be assumed to be uniform flow in order to remove dependence on the design variable D. 6. Compute flow variable sensitivities for the current time-level by iteratively solving the linear system: [ ] R n U n U n = Rn (62) 7. Perform vector-vector inner products as per equation (57) to determine the local objective sensitivity to the design variable D. Multiply this quantity by the global-to-local objective sensitivity and augment previously computed dl/dd 8. Continue egrating forward in time. At the end of the time-egration process, the complete sensitivity of the global objective L g to the design variable D is available. E. Adjo or Reverse Linearization For problems with multiple design variables and a single objective function, equation (57) is transposed to obtain the adjo or reverse linearization. Shape optimization problems require the adjo linearization to obtain sensitivities since there are typically numerous design variables and one or few objectives. he transpose of the forward linearization can be written as: [ ] dl n = dd { ( n=n f n= x n R n x n [ ] dl g = dd + xn R n n=n f n= x n. Case-A : Non-ime-Integrated Objective Function [ ] dl n [ ] dl g (63) dd + Un dl n R n U n ) [ ] R n L n U n U n } + xn L n x n (64) Consider the situation where the objective function is defined as a function of only the state variables and mesh coordinates at the end of the time-egration process. For such a case the derivative L n / n=n f is not just the local objective sensitivity to the design variables D at n = n f, but is also the final sensitivity required. he vectors L n / U n and L n / x n are easily computable since Ln is a scalar quantity. From equation (64) it is clear that the transpose of the inverse flow Jacobian matrix is required in order to proceed with the sensitivity computation. Since direct inversions of the flow Jacobian matrix are to be avoided due to cost reasons, a new variable called the flow adjo is defined as per equation (65) to enable an iterative approach: [ ] R Λ n n [ ] L n U = U n U n (65) (66) of 34

11 which can also be written as: [ R n U n ] [ ] L Λ n n U = U n (67) he flow adjo can be obtained by iteratively solving the linear system shown in equation (67). he solution convergence is similar to the convergence of any single linear system arising while solving the non-linear flow constra equation since both problems contain the same eigenvalues. Solving for the flow adjo and substituting in equation (64) results in three terms (A) n,(b) n and (C) n. he remaining preexisting contribution to Ln is denoted term (E) n. x n R n x n Λ n U } {{ } (A) n + xn R n x n Λ n U + Un R n U n Λ n U } {{ } (B) n } {{ } (C) n + xn L n x n } {{ } (E) n (68) erms (A) n, (B) n and (E) n can be expressed in terms of the surface displacement sensitivities of the deformed airfoil as: (A) n = x sur f (B) n = x sur f (E) n = x sur f x n sur f x n R n x sur f x n sur f x n x n sur f x sur f x n x n sur f R n x n x n sur f x n L n x sur f x n sur f x n Λ n U (69) Λ n U (7) he second quantity in the expanded versions of these terms are merely transposes of the corresponding rotation matrices [β]. his can be derived starting from equation (25) as: (7) δx n sur f = {[β] n [I]}x sur f (72) x n sur f = [β] n x sur f (73) x n sur f x sur f x n sur f x sur f = [β] n (74) = [β] n (75) he mesh motion constra equation provides the third quantity in these expressions. erms (A) n,(b) n and (E) n can now be written as: [K]δx = δx sur f (76) [K] x x sur f = [I] (77) x x sur f (A) n = x sur f (B) n = x sur f (E) n = x sur f = [K] (78) [β] n [K] R n x n Λ n U (79) [β] n [K] [β] n [K] L n x n R n x n Λ n U (8) (8) of 34

12 he computation of the second quantity in term (C) n is trivial since by examination of equation (52) it is obvious that U n occurs linearly. he product of the second and third quantities in term (C) n is denoted λ n U. In order to determine Un U n ( U n λ n U = he flow adjo variable Λ n U λ n U = Rn U n Λ n U = An Λ n U (82) t in term (C)n, a new expression that is analogous to equation (56) is written and transposed. [ ] R n ) ( R n x n = U n x n + Rn x n 2 x n 2 + Rn U n 2 U n 2 (83) x n R n x n + xn 2 R n x n 2 + Un 2 R n U n 2 for the time-level n can now be defined and solved as: ) [ R n U n ] λ n U (84) he final form of term (C) n is therefore: [ R n U n ] Λ n U = λn U (85) (C) n = xn R n x n Λ n U } {{ } (A) n + xn 2 R n Λ n U } x n 2 {{ } (B) n + Un 2 R n U n 2 Λ n U } {{ } (C) n (86) As in equation (8), (A) n from equation (86) can again be written in terms of the surface displacement sensitivity of the deformed airfoil as: (A) n = x sur f [β] n [K] R n x n Λ n U (87) Combining terms (A) n and (B) n a mesh motion adjo variable Λ n x is defined and solved for as shown in equation (88). ( ) ( ) [K] Λ n R n x = Λ n U + R n Λ n U (88) x n he mesh motion adjo variable can be iteratively solved for with cost similar to a mesh motion solution. he coefficient matrices again for both sets of equations are the same with the exception of the transpose. Note that although technically the combination of (A) n and (B) n to form a single mesh adjo is not required, this step reduces the number of mesh adjo solutions per time-level from two to one. he combined term (A) n + (B) n can now be computed as: (A) n + (B) n = x sur f x n [β] n Λ n x (89) A recurrence pattern is clearly seen from equation (86). U n 2 / in equation (86) will have to be solved for by substituting an expression analogous to equation (83) and defining an adjo variable Λ n 2 U. his will again yield a term (A) n 2 that can be combined with (B) n and a mesh adjo Λ n 2 x can be defined and solved. he procedure described above is simply an egration backward in time from time-level n = n f to the starting time-level n =. When the reverse time-egration reaches the first time-level, the recurrence phenomenon stops since the quantity U / is defined based on initial conditions or can be solved for by obtaining a steady-state adjo solution. Also, just as (B) n and (A) n were combined to define and solve the mesh adjo Λ n x, (A) n and (E) n are combined at the final time-level n to define and solve a mesh adjo Λ n x. { ( ) [K] L Λ n n ( ) } R n x = + Λ n U (9) x n x n 2 of 34

13 his can then be multiplied to get the final answer for the combination of (A) n and (E) n. (A) n + (E) n = x sur f Adding all four terms yields the final sensitivity vector: [β] n Λ n x (9) L n = (A)n + (B) n + (C) n + (E) n (92) he procedure for computing the sensitivity vector dd dl using the adjo approach is as follows:. Begin backward time-egration at n = n f. 2. Compute vectors Ln U n and Ln x n. 3. Solve for flow adjo Λ n U as: [ R n ] [ ] L U n Λ n n U = U n (93) 4. Solve for mesh adjo Λ n x as per equation (9) and multiply by surface displacement sensitivities of deformed airfoil and also by rotation matrix [β] n to get sum of (A) n and (E) n as per equation (9). 5. Multiply Λ n U by An t to get λ n U. 6. Solve for flow adjo at previous time-level n as: [ R n U n 7. Solve for mesh adjo Λ n x as per equation (88) and determine (B) n. ] Λ n U = λn U (94) 8. Recursively egrate backward in time all the way to the first time-level in order to assemble (C) n, solving for a flow adjo and a mesh adjo at each time-level. 9. At the end of the reverse egration process the complete sensitivity vector is available. 2. Case-B: ime-integrated Objective Function For a situation where the final objective function is obtained through egration in time or in discrete terms a function of some local objective computed at each time-level, it appears that the procedure described above would have to be repeated for each time-level to obtain dln dlg dd after which dd can be assembled. A slight modification in the derivation of the unsteady adjo can circumvent this tedious process. In the previous case the reverse time-egration was initiated at the final time-level and proceeded to the first time-level in order to compute the sensitivity of a global objective whose definition was based only on the solution and mesh coordinates at the final time-level. If the global objective L g is redefined as a quantity egrated in time, or as a quantity dependent on the solutions at each timelevel n = n f,n f,...2,,, the effect of these local sensitivities can be gathered in a single backward egration beginning at the final time-level, thus eliminating the need to evaluate these independently for each time-level n. Expanding equation (63) and reversing the summation order yields: dl g dd = dln dd dl g dl n + dln dd Further expanding the local objective sensitivities: ( ) dl g U n L n = dd U n + xn L n dl g x n dl n }{{} () + dl g dl n ( + dl dd U n L n + xn dl g dl L n ) dl g } U n {{ x n dl n } (Y ) (95) (96) 3 of 34

14 Case-A presented a method to compute term (). Equation (96) falls back to Case-A if the terms L g / L n for n = n f,... vanish and L g / L n f =. In order to avoid computing term (Y ) independently of term (), the derivation from Case-A is modified to include its effect. his applies to all terms recursively appearing in the expansion shown in equation (95). he modifications to the derivation are:. Firstly the flow adjo variable indicated in equation (65) is redefined as: [ ] R Λ n n [ ] L n [ ] dl g U = U n U n dl n (97) 2. (E) n in equation (8) is modified to include the global-to-local objective sensitivity: (E) n = x sur f [β] n [K] L n x n dl g dl n (98) 3. Previously term (C) n was defined as: (C) n = Un λ n U (99) A crucial difference in the definition of term (C) n is to be emphasized at this juncture. he basis of the previous derivation was the substitution of a suitable expression for the flow variable sensitivities U n / in order to proceed with the backward egration. While this remains true for Case-B, examination of term (Y ) in equation (96) indicates an additional contribution to λ n U from time-level n. Modifying the definition of λn U to include this contribution results in: λ n U = Rn U n Λ n U + Ln dl g U n dl n It is clear that although the egration was initiated at the final time-level n f, the effect of the local-to-global sensitivity from time-level n f has been included via the second term, thus eliminating the necessity to evaluate this independently. 4. Similarly the mesh adjo Λ n x is redefined such that it includes objective information from time-level n. he additional contribution in this case is due to the appearance of x n / in term (Y ) of equation (96). ( ) ( ) [K] Λ n R n x = Λ n U + R n ( ) L Λ n n ( ) dl g U + x n dl n () x n x n 5. he mesh adjo at the final time-level Λ n x is also changed to include the global-to-local objective relation. { ( ) [K] L Λ n n ( ) dl g ( ) } R n x = dl n + Λ n U (2) x n 6. Additionally, since there are local objectives computed at each time-level, the linearizations of these with respect to the mesh coordinates and flow solution at the respective time-levels are required. his translates o evaluating the vectors Ln U n and Ln x n at each time-level n. F. Modification of Formulation for BDF2 ime-integration Scheme he unique character of the derivation permits the application of the same technique regardless of the number of backward steps required in the time discretization. For a two-step backward difference formula, the constra equation has contributions from time indices n,n and n 2. he new constra equation written in functional form is therefore: x n () R n = F [ x n,x n,x n 2 U n,u n,u n 2] = (3) 4 of 34

15 he flow variable sensitivity at time-index n obtained through differentiation of equation (3) with respect to design variables D is: U n [ ] ( ) R n = R n x n U n x n + Rn x n x n + Rn x n 2 x n 2 + Rn U n U n + Rn U n 2 U n 2 (4) he linearization at any time-level n now has additional contributions from time-level n 2. For the BDF scheme, the unsteady adjo derivation resulted in a two-po recurrence relation in time involving the time indices n and n. For the BDF2 time egration scheme, the recurrence relation is based on three-pos in time, namely indices n, n and n 2. Equations () and () will now have additional source terms from time-index n 2. It is eresting to note that irrespective of the number of backward steps required in the time-egration scheme, only one flow adjo and one mesh adjo solution is required at each time-level. A. Code Structure V. Implementation Details he software is divided o three main components, namely the flow solver, the adjo/tangent solver, and the optimization script. he flow solver is an unstructured cell-centered unsteady finite-volume code with moving mesh capability that is spatially and temporally second-order accurate. he adjo and tangent solvers consist of linearizations of the physics routines used in the flow solver and iterative solution routines retained from the flow solver. he flow solver performs the time-egration process to compute an unsteady flow solution that is then written to disk. he adjo or tangent solver reads the solution from disk and computes the required sensitivities. A simple bash script is used as the optimization controller and calls the flow and adjo/tangent solvers. All data exchange between the flow and adjo/tangent solvers occur via files written to disk. his is necessary since the adjo solver performs a backward egration in time and thus requires the solution history. It would be impractical to hold the entire unsteady solution set in memory for later use by the adjo solver, however file I/O operations consume trivial amount of time. In contrast to the adjo solver, the tangent solver has the ability to compute sensitivities simultaneously with the flow solver but with a lag of one time-level. his can be done since the tangent solver also performs a forward egration in time and can discard old values as it proceeds forward in time. B. Linearization for 2 nd order Spatial Accuracy Computation of the flow adjo variable requires the linearization of the flow constra or flow residual equation with respect to the flow (state) variables. Such a linearization results in the flow Jacobian matrix R/ U and is used in the Newton solver employed to obtain the flow solution. In the case of a spatially first-order accurate discretization this results in a nearest neighbor stencil where the flow residual R for any element in the mesh is a function of its own state variables and the state variables of its immediate neighbors. On triangular unstructured meshes this translates o the flow Jacobian matrix being a sparse matrix with each row consisting of a dominant diagonal block element and three off-diagonal block elements. A simple edge-based data structure is sufficient for the purpose of constructing and storing the elements of the flow Jacobian matrix. In the case of second-order spatial accuracy the flow residual of each element is no longer restricted to a nearest neighbor stencil since the residual depends not only on the state variables but also their spatial gradients. he gradients are functions of erpolated nodal state values which in turn are functions of the states of elements that share that node. Constructing and storing an exact flow Jacobian matrix quickly becomes impractical due to large memory requirements and the lack of a simple data structure. For the flow solver, it is not required that an exact linearization of the flow residual be available since only the solution of R(U) = is necessary. In order to achieve second-order accuracy, it is only required that the flow residual itself be constructed using second-order extrapolations of U. An inexact Newton s method is employed where the first-order accurate flow Jacobian matrix is used in solving second-order accurate residual equations R(U). he method being inexact no longer exhibits the quadratic rate of convergence typical of a Newton solver but greatly simplifies the solution procedure. In the case of the adjo solver, it has been shown that approximating a second-order accurate flow Jacobian matrix with a first-order accurate matrix leads to significant errors in the computed sensitivities. 25 his becomes especially true in regions of the flow where second-order accuracy is important or necessary. he linear systems that involve the flow Jacobian matrix are equations (62) and (67). Although computing and storing the exact second-order Jacobian is impractical, it is quite straightforward to construct the product of the second-order Jacobian and a vector using a two-pass approach. 7, 8 his property is taken advantage of and a defect correction method is used to solve the linear 5 of 34

16 system involving the second-order flow Jacobian matrix. For example consider the left-hand-side of the linear system shown in equation (67). In the case where the flow Jacobian matrix is second-order accurate, this can be split o the sum of two matrix-vector products as follows: [ ] R {[ R Λ U = U 2 U [ ] R = ] Λ U + U }{{} () [ ][ ]} R U Λ U (5) U U [ ] U [ ] R + Λ U (6) U U }{{} (2) where [ R/ U] is the first-order Jacobian. erm () can be easily computed since the first-order Jacobian is already stored. erm (2) can be computed via a series of matrix-vector products on the fly if Λ U is available. he defect correction method is similar to a non-linear solution method and can mathematically be expressed as: { [ ] R [ ] } L [P]δΛU k = ΛU k (7) U 2 U Here [P] refers to the preconditioner and δλ U the correction for Λ U. When the correct value for Λ U has been achieved the right-hand-side of equation (7) goes to zero. In order for the system to converge, the preconditioner [P] must be closely related to the second-order flow Jacobian matrix. ypically the preconditioner is taken to be the transpose of the first-order flow Jacobian matrix. 26 he right-hand-side includes the effect of the second-order flow Jacobian matrix and is evaluated as described by equation (6). C. Linearization of Roe-Averaged Flux Jacobian Matrix he flux model used in the analysis code is based on Roe s approximate Riemann solver. ypically the flux Jacobian [Â] is assumed to be a constant during the flow residual linearization process, since in most cases its linearization has little effect on the overall accuracy of the flow Jacobian matrix R/ U. 25 Considering that prior work on the importance of including the exact linearization of the [Â] matrix was done in the context of steady-state problems, it was decided that its effect in unsteady flows must be studied. Several methods to include the effect of the [Â] matrix were investigated in the process. hese were:. Hybrid: finite-difference + linearization (FDL) of flux routines 2. Hybrid: complex variable differentiation + linearization (CVDL) of flux routines 3. Exact linearization he easiest method was to approach the problem from a hybrid standpo, where an exact linearization is used to compute all components of the flow Jacobian matrix except the [Â] matrix which is finite-differenced or differentiated using the complex variable method. he hybrid techniques are relatively easier in the sense that a tedious linearization of the dissipative flux model need not be undertaken. he disadvantage of using FDL or CVDL to obtain the linearization of the [Â] matrix is that it involves five or four function calls of the Roe flux routine respectively, rather than just one function call of the linearized routine. Additionally the step-size for the finite difference in FDL remains a variable, and it would be uneconomical to perform finite-differencing with several different step-sizes to choose a suitable value during the course of an optimization. he CVDL method is more involved than the FDL method since it requires the redefinition of variables used in functional evaluations as complex variables. he CVDL method also requires a step-size but is not as sensitive as the FDL method to the step-size itself. Details on the implementation of this method can be found in the following subsection. An exact linearization of the dissipative flux model was also performed but no comparisons in terms of cost were done. From an accuracy standpo it would appear that an exact linearization would be more appropriate since the choice of step-size for the hybrid methods remains a variable. D. Complex Variable Based Differentiation27, 28 Any function operator f (x) that operates on a real variable x can be utilized to compute the derivative f (x) by redefining the input variable x and all ermediate variables used in f (x) as a complex variables. For a complex input, 6 of 34

17 the function when redefined as described produces a complex output. he derivative of the real function f (x) can be computed by expanding the complex operator f (x + ih). he expansion of the complex operator can be written as: he derivative f (x) is then simply: f (x + ih) = f (x) + ih f (x) + (8) f (x) = Im[ f (x + ih)] h In the case of the finite-difference method the step-size is of extreme importance. A large step-size will lead to inaccuracies in the computed slope, while on the other end, too small a step-size will generate errors due to machine precision. here is usually a small range of step-sizes over which the computed slope values converge, but in several cases no such region can be identified. he complex variable method overcomes this challenge although it is similar to the finite-difference method in the sense that it also requires a step size h. he advantage over the finite-difference method is that it is not sensitive to extremely small step sizes. he complex variable method is useful in validating sections of the linearized code when there are significant discrepancies between the finite-differenced and adjobased values computed for the sensitivity. E. Optimization Procedure he optimization algorithm used in the example problems is the simple steepest-descent method. he optimization process can be broken down o the following steps:. Input the baseline or deformed geometry. 2. Call the flow solver to obtain an unsteady solution. 3. Evaluate the objective function based on the unsteady solution. 4. Call the adjo solver to obtain the sensitivity vector dl dd. 5. Compute the perturbation of the design variables using the sensitivity as: (9) δd = λ dl dd () where λ is the step size and the negative sign indicates the direction for minimizing the objective. 6. Deform the airfoil geometry using computed perturbation and the deformation function. 7. Repeat the process until an optimum solution is achieved. F. Linear Multigrid A linear geometric agglomeration multigrid 2 strategy is used to accelerate convergence of all linear systems encountered in the solution process, be either, the flow, mesh motion, flow adjo or mesh adjo equations. he linear multigrid method uses a correction scheme where coarser levels recursively provide corrections to the fine grid solution. he coarse level meshes are built by repeated agglomeration or merging of neighboring control volumes to form a single control volume. Figures (3(a)) through (3(f)) show an unstructured triangular mesh around a NACA2 airfoil and the agglomerated coarser levels. Consider a linear system developed by discretizing and linearizing a non-linear equation on the fine level mesh. [A] h x h = b h () An approximate solution to the linear system can be obtained by using conventional iterative methods such as Jacobi or Gauss-Seidel iterations. he residual of the linear system can then be written as: [A] h x h b h = r h (2) 7 of 34